1.
Johann Gottlieb Fichte
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Recently, philosophers and scholars have begun to appreciate Fichte as an important philosopher in his own right due to his original insights into the nature of self-consciousness or self-awareness. Fichte was also the originator of thesis–antithesis–synthesis, an idea that is often attributed to Hegel. Like Descartes and Kant before him, Fichte was motivated by the problem of subjectivity, Fichte also wrote works of political philosophy, he has a reputation as one of the fathers of German nationalism. Fichte was born in Rammenau, Upper Lusatia, the son of a ribbon weaver, he came of peasant stock which had lived in the region for many generations. The family was noted in the neighborhood for its probity and piety, christian Fichte, Johann Gottliebs father, married somewhat above his station. It has been suggested that a certain impatience which Fichte himself displayed throughout his life was an inheritance from his mother, young Fichte received the rudiments of his education from his father. He early showed remarkable ability, and it was owing to his reputation among the villagers that he gained the opportunity for an education than he otherwise would have received. The story runs that the Freiherr von Militz, a country landowner and he was, however, informed that a lad in the neighborhood would be able to repeat the sermon practically verbatim. As a result, the baron took the lad into his protection, Fichte was placed in the family of Pastor Krebel at Niederau near Meissen and there received thorough grounding in the classics. From this time onward, Fichte saw little of his parents, in October 1774, he was attending the celebrated foundation-school at Pforta near Naumburg. This school is associated with the names of Novalis, August Wilhelm Schlegel, Friedrich Schlegel, perhaps his education strengthened a tendency toward introspection and independence, characteristics which appear strongly in his doctrines and writings. In 1780, he study at the theology seminary of University of Jena. He was transferred a year later to study at the Leipzig University, Fichte seems to have supported himself at this period of bitter poverty and hard struggle. Freiherr von Militz continued to him, but when he died in 1784, Fichte had to end his studies prematurely. During the years 1784 to 1788, he supported himself in a way as tutor in various Saxon families. In early 1788, he returned to Leipzig in the hope of finding a better employment and he lived in Zurich for the next two years, which was a time of great contentment for him. There he met his wife, Johanna Rahn, and Johann Heinrich Pestalozzi. There he also became in 1793 a member of the Freemasonry lodge Modestia cum Libertate with which Johann Wolfgang Goethe was also connected, in the spring of 1790, he became engaged to Johanna

2.
Scholastik
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It originated as an outgrowth of and a departure from Christian monastic schools at the earliest European universities. The Scholastic thought is known for rigorous conceptual analysis and the careful drawing of distinctions. Because of its emphasis on rigorous dialectical method, scholasticism was eventually applied to other fields of study. Some of the figures of scholasticism include Anselm of Canterbury, Peter Abelard, Alexander of Hales, Albertus Magnus, Duns Scotus, William of Ockham, Bonaventure. Important work in the tradition has been carried on well past Aquinass time, for instance by Francisco Suárez and Luis de Molina. The terms scholastic and scholasticism derive from the Latin word scholasticus and the latter from the Greek σχολαστικός, forerunners of Christian scholasticism were Islamic Ilm al-Kalām, literally science of discourse, and Jewish philosophy, especially Jewish Kalam. The first significant renewal of learning in the West came with the Carolingian Renaissance of the Early Middle Ages, charlemagne, advised by Peter of Pisa and Alcuin of York, attracted the scholars of England and Ireland. By decree in AD787, he established schools in every abbey in his empire and these schools, from which the name scholasticism is derived, became centers of medieval learning. During this period, knowledge of Ancient Greek had vanished in the west except in Ireland, Irish scholars had a considerable presence in the Frankish court, where they were renowned for their learning. Among them was Johannes Scotus Eriugena, one of the founders of scholasticism, Eriugena was the most significant Irish intellectual of the early monastic period and an outstanding philosopher in terms of originality. He had considerable familiarity with the Greek language and translated works into Latin, affording access to the Cappadocian Fathers. The other three founders of scholasticism were the 11th-century scholars Peter Abelard, Archbishop Lanfranc of Canterbury and Archbishop Anselm of Canterbury and this period saw the beginning of the rediscovery of many Greek works which had been lost to the Latin West. As early as the 10th century, scholars in Spain had begun to gather translated texts and, in the half of that century. After a successful burst of Reconquista in the 12th century, Spain opened even further for Christian scholars, as these Europeans encountered Islamic philosophy, they opened a wealth of Arab knowledge of mathematics and astronomy. Scholars such as Adelard of Bath traveled to Spain and Sicily, translating works on astronomy and mathematics, at the same time, Anselm of Laon systematized the production of the gloss on Scripture, followed by the rise to prominence of dialectic in the work of Abelard. Peter Lombard produced a collection of Sentences, or opinions of the Church Fathers and other authorities The 13th, the early 13th century witnessed the culmination of the recovery of Greek philosophy. Schools of translation grew up in Italy and Sicily, and eventually in the rest of Europe, powerful Norman kings gathered men of knowledge from Italy and other areas into their courts as a sign of their prestige. His work formed the basis of the commentaries that followed

3.
Kontradiktion
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In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotles law of noncontradiction states that One cannot say of something that it is, by extension, outside of classical logic, one can speak of contradictions between actions when one presumes that their motives contradict each other. By creation of a paradox, Platos Euthydemus dialogue demonstrates the need for the notion of contradiction, in the ensuing dialogue Dionysodorus denies the existence of contradiction, all the while that Socrates is contradicting him. I in my astonishment said, What do you mean Dionysodorus, the dictum is that there is no such thing as a falsehood, a man must either say what is true or say nothing. Indeed, Dionysodorus agrees that there is no such thing as false opinion, there is no such thing as ignorance and demands of Socrates to Refute me. Socrates responds But how can I refute you, if, as you say, note, The symbol ⊥ represents an arbitrary contradiction, with the dual tee symbol ⊤ used to denote an arbitrary tautology. Contradiction is sometimes symbolized by Opq, and tautology by Vpq, the turnstile symbol, ⊢ is often read as yields or proves. In classical logic, particularly in propositional and first-order logic, a proposition φ is a contradiction if, since for contradictory φ it is true that ⊢ φ → ψ for all ψ, one may prove any proposition from a set of axioms which contains contradictions. This is called the principle of explosion or ex falso quodlibet, in a complete logic, a formula is contradictory if and only if it is unsatisfiable. Therefore, a proof that ¬ φ ⊢ ⊥ also proves that φ is true, the use of this fact constitutes the technique of the proof by contradiction, which mathematicians use extensively. This applies only in a logic using the excluded middle A ∨ ¬ A as an axiom, in mathematics, the symbol used to represent a contradiction within a proof varies. A consistency proof requires an axiomatic system a demonstration that it is not the case both the formula p and its negation ~p can be derived in the system. Posts solution to the problem is described in the demonstration An Example of a Successful Absolute Proof of Consistency offered by Ernest Nagel and they too observe a problem with respect to the notion of contradiction with its usual truth values of truth and falsity. They observe that, The property of being a tautology has been defined in notions of truth, yet these notions obviously involve a reference to something outside the formula calculus. Therefore, the mentioned in the text in effect offers an interpretation of the calculus. This being so, the authors have not done what they promised, namely, proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem. Given some primitive formulas such as PMs primitives S1 V S2, so what will be the definition of tautologous

4.
Negation
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Negation is thus a unary logical connective. It may be applied as an operation on propositions, truth values, in classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p. Classical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true. So, if statement A is true, then ¬A would therefore be false, the truth table of ¬p is as follows, Classical negation can be defined in terms of other logical operations. For example, ¬p can be defined as p → F, conversely, one can define F as p & ¬p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false, while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. But in classical logic, we get an identity, p → q can be defined as ¬p ∨ q. Algebraically, classical negation corresponds to complementation in a Boolean algebra and these algebras provide a semantics for classical and intuitionistic logic respectively. The negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following, In set theory \ is also used to indicate not member of, U \ A is the set of all members of U that are not members of A. No matter how it is notated or symbolized, the negation ¬p / −p can be read as it is not the case p, not that p. Within a system of logic, double negation, that is. In intuitionistic logic, a proposition implies its double negation but not conversely and this marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two and this result is known as Glivenkos theorem. De Morgans laws provide a way of distributing negation over disjunction and conjunction, ¬ ≡, in Boolean algebra, a linear function is one such that, If there exists a0, a1. An ∈ such that f = a0 ⊕ ⊕, another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference. Negation is a logical operator

5.
Aussagenlogik
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Logical connectives are found in natural languages. In English for example, some examples are and, or, not”, the following is an example of a very simple inference within the scope of propositional logic, Premise 1, If its raining then its cloudy. Both premises and the conclusion are propositions, the premises are taken for granted and then with the application of modus ponens the conclusion follows. Not only that, but they will also correspond with any other inference of this form, Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions, a constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly, usually in truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false. Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic, although propositional logic had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics. The logic was focused on propositions and this advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood, consequently, the system was essentially reinvented by Peter Abelard in the 12th century. Propositional logic was eventually refined using symbolic logic, the 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community, consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, one author describes predicate logic as combining the distinctive features of syllogistic logic and propositional logic. Consequently, predicate logic ushered in a new era in history, however, advances in propositional logic were still made after Frege, including Natural Deduction. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz, Truth-Trees were invented by Evert Willem Beth. The invention of truth-tables, however, is of controversial attribution, within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure, itself, is credited to either Ludwig Wittgenstein or Emil Post

6.
Bezugssystem
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In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements. In n dimensions, n+1 reference points are sufficient to define a reference frame. Using rectangular coordinates, a frame may be defined with a reference point at the origin. In Einsteinian relativity, reference frames are used to specify the relationship between an observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes observational frame of reference, a relativistic reference frame includes the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent, the need to distinguish between the various meanings of frame of reference has led to a variety of terms. For example, sometimes the type of system is attached as a modifier. Sometimes the state of motion is emphasized, as in rotating frame of reference, sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference, in this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a system may be employed for many purposes where the state of motion is not the primary concern. For example, a system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors and it seems useful to divorce the various aspects of a reference frame for the discussion below. A coordinate system is a concept, amounting to a choice of language used to describe observations. Consequently, an observer in a frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. A change in the choice of coordinate system does not change an observers state of motion. This viewpoint can be found elsewhere as well, which is not to dispute that some coordinate systems may be a better choice for some observations than are others. Choice of what to measure and with what observational apparatus is a separate from the observers state of motion. D. Norton, The discussion is taken beyond simple space-time coordinate systems by Brading, extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity

7.
Satz vom Widerspruch
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This article uses forms of logical notation. For a concise description of the used in this notation. In classical logic, the law of non-contradiction is the second of the three laws of thought. It states that contradictory statements cannot both be true in the same sense at the time, e. g. the two propositions A is B and A is not B are mutually exclusive. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as, the law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle, an expression of its jointly exhaustive aspect. One difficulty in applying the law of non-contradiction is ambiguity in the propositions, for instance, if time is not explicitly specified as part of the propositions A and B, then A may be B at one time, and not at another. A and B may in some cases be made to sound mutually exclusive linguistically even though A may be partly B and partly not B at the same time. However, it is impossible to predicate of the thing, at the same time, and in the same sense, the absence. According to both Plato and Aristotle, Heraclitus was said to have denied the law of non-contradiction and this is quite likely if, as Plato pointed out, the law of non-contradiction does not hold for changing things in the world. If a philosophy of Becoming is not possible without change, then what is to become must already exist in the present object, unfortunately, so little remains of Heraclitus aphorisms that not much about his philosophy can be said with certainty. The road up and down are one and the same implies either the road leads both ways, or there can be no road at all and this is the logical complement of the law of non-contradiction. According to Heraclitus, change, and the constant conflict of opposites is the logos of nature. Personal subjective perceptions or judgments can only be said to be true at the time in the same respect, in which case. The most famous saying of Protagoras is, Man is the measure of all things, of things which are, that they are, and of things which are not, however, Protagoras was referring to things that are used by or in some way related to humans. This makes a difference in the meaning of his aphorism. Properties, social entities, ideas, feelings, judgements, etc. originate in the human mind, however, Protagoras has never suggested that man must be the measure of stars, or the motion of the stars. Parmenides employed an ontological version of the law of non-contradiction to prove that being is and to deny the void, change and he also similarly disproved contrary propositions. In his poem On Nature, he said, The nature of the ‘is’ or what-is in Parmenides is a contentious subject

8.
Disjunktion
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In logic and mathematics, or is the truth-functional operator of disjunction, also known as alternation, the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is written as ∨ or +. A or B is true if A is true, or if B is true, or if both A and B are true. In logic, or by means the inclusive or, distinguished from an exclusive or. An operand of a disjunction is called a disjunct, related concepts in other fields are, In natural language, the coordinating conjunction or. In programming languages, the short-circuit or control structure, or is usually expressed with an infix operator, in mathematics and logic, ∨, in electronics, +, and in most programming languages, |, ||, or or. In Jan Łukasiewiczs prefix notation for logic, the operator is A, logical disjunction is an operation on two logical values, typically the values of two propositions, that has a value of false if and only if both of its operands are false. More generally, a disjunction is a formula that can have one or more literals separated only by ors. A single literal is often considered to be a degenerate disjunction, the disjunctive identity is false, which is to say that the or of an expression with false has the same value as the original expression. In keeping with the concept of truth, when disjunction is defined as an operator or function of arbitrary arity. Falsehood-preserving, The interpretation under which all variables are assigned a value of false produces a truth value of false as a result of disjunction. The mathematical symbol for logical disjunction varies in the literature, in addition to the word or, and the formula Apq, the symbol ∨, deriving from the Latin word vel is commonly used for disjunction. For example, A ∨ B is read as A or B, such a disjunction is false if both A and B are false. In all other cases it is true, all of the following are disjunctions, A ∨ B ¬ A ∨ B A ∨ ¬ B ∨ ¬ C ∨ D ∨ ¬ E. The corresponding operation in set theory is the set-theoretic union, operators corresponding to logical disjunction exist in most programming languages. Disjunction is often used for bitwise operations, for example, x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged. Logical disjunction is usually short-circuited, that is, if the first operand evaluates to true then the second operand is not evaluated, the logical disjunction operator thus usually constitutes a sequence point. In a parallel language, it is possible to both sides, they are evaluated in parallel, and if one terminates with value true

9.
Realität
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Reality is the state of things as they actually exist, rather than as they may appear or might be imagined. Reality includes everything that is and has been, whether or not it is observable or comprehensible, a still broader definition includes that which has existed, exists, or will exist. By contrast existence is restricted solely to that which has physical existence or has a direct basis in it in the way that thoughts do in the brain. Reality is often contrasted with what is imaginary, illusory, delusional, in the mind, dreams, what is false, what is fictional, at the same time, what is abstract plays a role both in everyday life and in academic research. For instance, causality, virtue, life, and distributive justice are abstract concepts that can be difficult to define, but they are only rarely equated with pure delusions. Both the existence and reality of abstractions are in dispute, one extreme position regards them as mere words and this disagreement is the basis of the philosophical problem of universals. The truth refers to what is real, while falsity refers to what is not, a common colloquial usage would have reality mean perceptions, beliefs, and attitudes toward reality, as in My reality is not your reality. This is often used just as a colloquialism indicating that the parties to a conversation agree, or should agree, for example, in a religious discussion between friends, one might say, You might disagree, but in my reality, everyone goes to heaven. Reality can be defined in a way that links it to world views or parts of them, Reality is the totality of all things, structures, events and phenomena and it is what a world view ultimately attempts to describe or map. Certain ideas from physics, philosophy, sociology, literary criticism, one such belief is that there simply and literally is no reality beyond the perceptions or beliefs we each have about reality. Many of the concepts of science and philosophy are often defined culturally and socially and this idea was elaborated by Thomas Kuhn in his book The Structure of Scientific Revolutions. The Social Construction of Reality, a book about the sociology of knowledge written by Peter L. Berger and it explained how knowledge is acquired and used for the comprehension of reality. Out of all the realities, the reality of life is the most important one since our consciousness requires us to be completely aware. Philosophy addresses two different aspects of the topic of reality, the nature of reality itself, and the relationship between the mind and reality. On the one hand, ontology is the study of being, and the topic of the field is couched, variously, in terms of being, existence, what is. The task in ontology is to describe the most general categories of reality, if a philosopher wanted to proffer a positive definition of the concept reality, it would be done under this heading. As explained above, some philosophers draw a distinction between reality and existence, in fact, many analytic philosophers today tend to avoid the term real and reality in discussing ontological issues. But for those who would treat is real the same way they treat exists and it has been widely held by analytic philosophers that it is not a property at all, though this view has lost some ground in recent decades

10.
Identität
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In philosophy, the matter of personal identity deals with such questions as, What makes it true that a person at one time is the same thing as a person at another time. Or What kinds of things are we persons, the term identity in personal identity refers to numerical identity, where saying that X and Y are numerically identical just means that X and Y are the same thing. Personal identity is not the same as personality, though some theories of personal identity maintain that continuity of personality may be required for one to persist through time. Many people claim we are animals, or organisms, but many others believe that no person can exist without mental traits. Since an organism can exist without consciousness, both these views cannot be true, thus, in order to determine whether certain features are crucial to a persons continued existence, it may be important to first ask what sort of things we are. Generally, personal identity is the unique identity of a person in the course of time. The synchronic problem concerns the question of, What features and traits characterize a person at a given time, in Continental philosophy and in Analytic philosophy, enquiry to the nature of Identity is common. Continental philosophy deals with conceptually maintaining identity when confronted by different philosophic propositions, postulates, one concept of personal persistence over time is simply to have continuous bodily existence. With humans, over time our bodies age and grow, losing and gaining matter and it is thus problematic to ground persistence of personal identity over time in the continuous existence of our bodies. Nevertheless, this approach has its supporters which define humans as a biological organism and this personal identity ontology assumes the relational theory of life-sustaining processes instead of bodily continuity. Derek Parfit presents an experiment designed to bring out intuitions about the corporeal continuity. This thought experiment discusses cases in which a person is teletransported from Earth to Mars, the mind-body problem concerns the explanation of the relationship, if any, that exists between minds, or mental processes, and bodily states or processes. One of the aims of philosophers who work in area is to explain how a non-material mind can influence a material body. However, this is not uncontroversial or unproblematic, and adopting it as a solution raises questions, perceptual experiences depend on stimuli which arrive at various sensory organs from the external world and these stimuli cause changes in mental states, ultimately causing sensation. A desire for food, for example, will tend to cause a person to move their body in a manner, the question, then, is how it can be possible for conscious experiences to arise out of an organ possessing electrochemical properties. A related problem is to explain how propositional attitudes can cause neurons of the brain to fire and these comprise some of the puzzles that have confronted epistemologists and philosophers of mind from at least the time of René Descartes. John Locke considered personal identity to be founded on consciousness, through this identification, moral responsibility could be attributed to the subject and punishment and guilt could be justified, as critics such as Nietzsche would point out. According to Locke, personal identity depends on consciousness, not on substance nor on the soul and we are the same person to the extent that we are conscious of the past and future thoughts and actions in the same way as we are conscious of present thoughts and actions

1.
An observer O, situated at the origin of a local set of coordinates – a frame of reference F. The observer in this frame uses the coordinates (x, y, z, t) to describe a spacetime event, shown as a star.

1.
The six-part fugue from The Musical Offering, in the hand of Johann Sebastian Bach

2.
Example of a tonal answer in J.S. Bach's Fugue No. 16 in G Minor, BWV 861, from the Well-Tempered Clavier, Book 1. (Listen) The first note of the subject, D (in red), is a prominent dominant note, demanding that the first note of the answer (in blue) sounds as the tonic, G.

3.
The interval of a fifth inverts to a fourth (dissonant) and therefore cannot be employed in invertible counterpoint, without preparation and resolution.